Multiscale asymptotic solutions to the Boltzmann equation for aerospace applications Thierry MAGIN Aeronautics and Aerospace Department von Karman Institute for Fluid Dynamics, Belgium Issues in Solving the Boltzmann Equation for Aerospace Applications ICERM Topical Workshop Brown University, Providence, June 3-7, 2013 Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 1 / 59
Acknowledgement Acknowledgement THANK YOU! Workshop organizers for this invitation to ICERM Collaborators who contributed to the results presented here Alessandro Munaf` o, Erik Torres and JB Scoggins (VKI) Benjamin Graille (Paris-Sud Orsay) Marc Massot (Ecole Centrale Paris) Vincent Giovangigli (Ecole Polytechnique) Irene Gamba and Jeff Haack (The University of Texas at Austin) Marco Panesi (University of Illinois at Urbana-Champaign) Rich Jaffe, David Schwenke, Winifred Huo (NASA ARC) Support from the E uropean R esearch C ouncil through Starting Grant #259354 Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 2 / 59
Acknowledgement Outline 1 Introduction 2 Translational thermal nonequilibrium in plasmas 3 Atomic ionization reactions 4 Internal energy excitation in molecular gases 5 Conclusion Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 3 / 59
Introduction Motivation Motivation: new challenges for aerospace science Design of spacecraft heat shields Modeling of the convective and radiative heat fluxes for: Robotic missions aiming at bringing back samples to Earth Manned exploration program to the Moon and Mars Intermediate eXperimental Vehicle of ESA NASA Mars Science Laboratory Hypersonic cruise vehicles Modeling of flows from continuum to rarefied conditions for the next generation of air breathing hypersonic vehicles Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 4 / 59
Introduction Motivation Engineering design in hypersonics Two quantities of interest relevant to rocket scientists Heat flux Shear stress to the vehicle surface ⇒ Complex multiscale problem Chemical nonequilibrium (gas) Dissociation, ionization, . . . Internal energy excitation Thermal nonequilibrium Blast capsule flow simulation Translational and internal energy relaxation VKI COOLFluiD platform Radiation and Mutation library Gas / surface interaction Surface catalysis Ablation Rarefied gas effects Turbulence (transition) Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 5 / 59
Introduction Objective Physico-chemical models for atmospheric entry plasmas Earth atmosphere: S = { N 2 , O 2 , NO , N , O , NO + , N + , O + , e − , . . . } Fluid dynamics Kinetic theory ρ i ( x , t ), i ∈ S, v ( x , t ), E ( x , t ) f i ( x , c i , t ) , i ∈ S Fluid dynamical description Gas modeled as a continuum in terms of macroscopic variables e . g . Navier-Stokes eqs., Boltzmann moment systems Kinetic description Gas particles of species i ∈ S follow a velocity distribution f i in the phase space ( x , c i ) e . g . Boltzmann eq. ⇒ Constraint: descriptions with consistent physico-chemical models Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 6 / 59
Introduction Objective From microscopic to macroscopic quantities Mass density of species i ∈ S: R ρ i ( x , t ) = f i m i d c i Mixture mass density: ρ ( x , t ) = P j ∈ S ρ j ( x , t ) Hydrodynamic velocity: R ρ ( x , t ) v ( x , t ) = P f j m j c j d c j j ∈ S Total energy (point particles): E ( x , t ) = P R 2 m j | c j | 2 d c j j 1 f j ∈ S Thermal (translational) energy: Velocity distribution function for 1D Ar ρ ( x , t ) e ( x , t ) = P R 2 m j | c j − v | 2 d c j j 1 f shockwave (Mach 3.38) at different j ∈ S positions x ∈ [ − 1 cm , +1 cm ] [Munafo et al. 2013] ⇒ Suitable asymptotic solutions can be derived by means of the Chapman-Enskog perturbative solution method Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 7 / 59
Introduction Objective Objective of this presentation “Engineers use knowledge primarily to design, produce, and operate artifacts . . . Scientists, by contrast, use knowledge primarily to generate more knowledge.” Walter Vincenti Derive asymptotic (hydrodynamic) solutions for atmospheric entry plasma flows from multiscale kinetic equations with entangled collision operators 1 Translational thermal nonequilibrium in plasmas and electro-magnetic field influence 2 Atomic ionization reactions Internal energy excitation in molecular gases 3 ⇒ Enrich mathematical models by adding more physics ⇒ Derive mathematical structure and fix ad-hoc terms found in engineering models ⇒ Integrate quantum chemistry databases “The numerical integration of the Boltzmann equation is a matter of national prestige . . . ” attributed to Anatoly Dorodnitsyn Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 8 / 59
Translational thermal nonequilibrium in plasmas 1. Translational thermal nonequilibrium in plasmas and electro-magnetic field influence [Magin, Graille, Massot, AIAA 2008] [Magin, Graille, Massot, NASA/TM-214578 2008] [Graille, Magin, Massot, M3AS 2009] Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 9 / 59
Translational thermal nonequilibrium in plasmas Assumptions Boltzmann equation Grad-Boltzmann limit: huge number of particles of very small diameter interacting with a finite mean free path value N 0 particles of characteristic diameter d 0 in a box of size L 0 Ratio of the olume occupied by these particles and the volume of the box: 1 6 π N 0 ( d 0 / L 0 ) 3 Limits N 0 → ∞ and d 0 / L 0 → 0 for finite 1 / Kn = N 0 π ( d 0 / L 0 ) 2 This volume ratio can be expressed as 1 6 ( d 0 / L 0 )[ N 0 π ( d 0 / L 0 ) 2 ] = 1 6 ( d 0 / L 0 ) / Kn This quantity tends to zero for a fixed value of the Kn number ⇒ the gas is dilute Assumptions S species: e . g . i ∈ S = { N 2 , O 2 , NO , N , O , N + 2 , O + 2 , NO + , N + , O + , e − } Dilute gas of n Point particles of mass m i : no internal energy Binary and elastic collisions (no chemical reactions) Molecular chaos : probabilities of finding particle i at point ( x , c i ) and particle j at point ( x , c j ) in the phase space are independent Collision pairs of arbitrary impact parameter are equiprobable : quantity f i ( x , c i ) does not change over distances of the order of the collision cross section Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 10 / 59
Translational thermal nonequilibrium in plasmas Assumptions Translational thermal nonequilibrium and electromagnetic field influence in multicomponent plasma flows Plasma composed of electrons (index e ), and heavy particles , atoms and molecules, neutral or ionized (set of indices H); the full mixture of species is denoted by the set S = { e } ∪ H h ) 1 / 2 ≪ 1 Scaling parameter: ε = ( m 0 e / m 0 1 Classical mechanics description provided that (m 0 h k B T 0 ) 1 / 2 k B T 0 1 ≪ c 2 and (n 0 ) 1 / 3 ≫ m 0 h P e Binary charged interactions with screening of the Coulomb potential 2 Λ ≃ n 0 4 3 πλ 3 Debye ≫ 1 e Reference electrical and thermal energies of the system are of the same order 3 q 0 E 0 L 0 ≃ k B T 0 4 Magnetic field influence determined by the Hall parameter magnitude b β e = q 0 B 0 e t 0 e = ε 1 − b ( b < 0, b = 0, b = 1) m 0 5 Continuum description for compressible flows: O ( M h ) ≫ ε Kn M h ≃ ε Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 11 / 59
Translational thermal nonequilibrium in plasmas Dimensional analysis Dimensional analysis of the Boltzmann eq. [Petit, Darrozes 1975] 2 thermal speeds � � � k B T 0 k B T 0 m 0 V 0 V 0 = ε V 0 e e = , h = e , ε = m 0 m 0 m 0 e h h 2 kinetic temporal scales e = l 0 h = l 0 = t 0 1 l 0 = t 0 t 0 e , with V 0 V 0 n 0 σ 0 ε e h 1 macroscopic temporal scale V 0 t 0 t 0 = L 0 v 0 = L 0 l 0 v 0 = 1 1 h h Knt 0 = h V 0 l 0 ε M h h Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 12 / 59
Translational thermal nonequilibrium in plasmas Dimensional analysis Change of variable: heavy-particle velocity frame [M3AS 2009] The peculiar velocities are given by the relations C e = c e − ε M h v h , C i = c i − M h v h , i ∈ H ⇒ The heavy-particle diffusion flux vanishes � � m j C j d C j = 0 j f j ∈ H The choice of the heavy-particle velocity frame v h is natural for plasmas. In this frame: Heavy particles thermalize All particles diffuse Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 13 / 59
Translational thermal nonequilibrium in plasmas Dimensional analysis Boltzmann equation: nondimensional form and scaling Electrons: e ε − b 1 � � ∂ t f e + ε M h ( C e + ε M h v h ) · ∂ x f e + M h Knq e ( C e + ε M h v h ) ∧ B · ∂ C e f e D v h 1 1 � � + ε M h q e E − ε M h · ∂ C e f e − ( ∂ C e f e ⊗ C e ): ∂ x v h = ε M h Kn J e D t Heavy particles: i ∈ H i + ε 2 − b q i 1 � � ∂ t f i + M h ( C i + M h v h ) · ∂ x f ( C i + M h v h ) ∧ B · ∂ C i f i m i M h Kn � 1 D v h q i 1 � + m i E − M h · ∂ C i f i − ( ∂ C i f i ⊗ C i ): ∂ x v h = M h Kn J i M h D t ⇒ The multiscale analysis ( ε, Kn , β e ) occurs at three levels in the kinetic eqs. in the crossed collision operators in the collisional invariants Thierry MAGIN (VKI) Multiscale asymptotic solutions 3 June 2013 14 / 59
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